Eigenvalue estimates for a three-dimensional magnetic Schr\"odinger operator

We consider a magnetic Schr"odinger operator $H^h=(-ih\mathrm{\nabla }-\vec{A}{)}^2$ with the Dirichlet boundary conditions in an open set $\Omega \subset {\mathbb R}^3$,where$h>0$isasmallparameter\mathrm{.}Wesupposethattheminimalvalue$b_0$ofthemodule$|\vec{B}|$ofthevectormagneticfield$\vec{B}$ is strictly positive, and there exists a unique minimum point of $\mathrm{\mid }\vec{B}\mathrm{\mid }$, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.